We show that the measure of non-compactness of the limiting embedding of Sobolev–Lorentz spaces is equal to the norm. This is a consequence of our general theorem for arbitrary Banach spaces.

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Let Ω ⊂ R n be an open set and let f ∈ W 1,p (Ω, R n ) be a weak (sequential) limit of Sobolev homeomorphisms. Then f is injective almost everywhere for p > n − 1 both in the image and in the domain. For p ≤ n − 1 we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and a topological image of a point is a continuum for every point in a set of positive measure in the domain.

Let Ω \Omega be a simply connected domain in C \mathbb {C} and let 0 > p > ∞ 0>p>\infty . We show that there is a quasiconformal mapping f f from the unit disk D \mathbb {D} onto Ω \Omega which is in the Hardy space H p H^p . We furthermore show that either all quasiconformal mappings from D \mathbb {D} onto Ω \Omega are in  H p H^p for every p p , or for every 0 > p > ∞ 0>p>\infty there is a quasiconformal mapping f : D → Ω f\colon \mathbb {D}\to \Omega with f ∉ H p f\notin H^p .